Skip to main content

putting 12 matchsticks end to end, exactly 3 different triangles can be made, what is the formula needed to calculate the number of triangles can...

There are three distinct triangles using 12 matchsticks: 3-4-5,4-4-4, and 5-5-2.

Note that for a given number of matchsticks M, if the sides of the triangle are labeled a,b, and c (without loss of generality let c be the longest side) then a+b+c=M, a+b>c and c<M/2.


We can generate the number of triangles for a given M:


For 24 matchsticks the possible triangles are 11-11-2,11-10-3,11-9-4,11-8-5,11-7-6,10-10-4,10-9-5,10-8-6,10-7-7,9-9-6,9-8-7,8-8-8 or 12 possible triangles.


There are 19 possible triangles for M=30.


For M>=3 then (M,number of triangles) is (3,1),(4,0),(5,1),(6,1),(7,2),(8,1),(9,3),(10,2),(11,4),(12,3),(13,5),(14,4),(15,7),(16,5),(17,8),(18,7)...


There is a formula for this:


If M is even then the number of triangles is the integer closest to M^2/48.


If M is odd then the number of triangles is the integer closest to (M+3)^2/48


For example, if M=12 we have 12^2/48=3.


If M=14 we have 14^2/48=4.083333 so there are 4 triangles.


If M=9 we have (9+3)^2/48=3.


If M=11 we have (11+3)^2/48=4.083333 so there are 4 triangles.


Please see the reference link for a proof.

Comments

Post a Comment

Popular posts from this blog

What is the meaning of "juggling fiends" in Macbeth?

Macbeth is beginning to realize that the three witches have been deceiving him since he first encountered them. Like jugglers, they have kept changing their forecasts in order create confusion. This is particularly apparent when the Second Apparition they raise in Act IV,   Scene 1 tells him that no man of woman born can overcome him in hand-to-hand battle--and then Macbeth finds himself confronted by the one man he has been avoiding out of a... Macbeth is beginning to realize that the three witches have been deceiving him since he first encountered them. Like jugglers, they have kept changing their forecasts in order create confusion. This is particularly apparent when the Second Apparition they raise in Act IV,   Scene 1 tells him that no man of woman born can overcome him in hand-to-hand battle--and then Macbeth finds himself confronted by the one man he has been avoiding out of a sense of guilt, and that man tells him: Despair thy charm. And let the angel whom thou still hast serve...
Post a Comment
Read more

In Martel's Life of Pi, how does writing a diary help Pi Patel build on his leadership skills?

At the end of chapter 73, Pi first divulges the fact that he had started a diary after about a week alone at sea. Pi claims that he wrote about "practical stuff" (208) such as what he caught for food, foreseeable problems and solutions, Richard Parker, and reflections about specific events that occurred. In a sense, Pi is the captain of his ship; therefore, he must take responsibility for himself and his crew. He can... At the end of chapter 73, Pi first divulges the fact that he had started a diary after about a week alone at sea. Pi claims that he wrote about "practical stuff" (208) such as what he caught for food, foreseeable problems and solutions, Richard Parker, and reflections about specific events that occurred. In a sense, Pi is the captain of his ship; therefore, he must take responsibility for himself and his crew. He can do this by writing down events, problems and solutions, and reflections to look back on when confronted with repeated problems. It can ...
Post a Comment
Read more

Why did Tybalt attack and kill Mercutio instead of Romeo, who he had originally been targeting?

Though Tybalt's quarrel is with Romeo, he begins the momentous duel in Act 3 by fighting with and killing Mercutio instead. Tybalt ultimately attacks Mercutio because the garrulous and hot-headed character insults Tybalt and goads him into a duel to protect Romeo's honor. From the beginning of the encounter in Act 3, Scene 1, it's clear that Mercutio is itching for a fight. For instance, when Tybalt signals that he wants to talk to Mercutio... Though Tybalt's quarrel is with Romeo, he begins the momentous duel in Act 3 by fighting with and killing Mercutio instead. Tybalt ultimately attacks Mercutio because the garrulous and hot-headed character insults Tybalt and goads him into a duel to protect Romeo's honor. From the beginning of the encounter in Act 3, Scene 1, it's clear that Mercutio is itching for a fight. For instance, when Tybalt signals that he wants to talk to Mercutio and Benvolio, Mercutio responds "And but one word with one of us?/ Couple it w...
Post a Comment
Read more